There are quite a few species of kestrel around the world, but I believe they are all quite similar in behaviour as well as in appearance -- however I can really only speak about the European kestrel, Falco tinnunculus.
It hunts mainly by hovering over open ground, tail fanned and facing the wind (giving it its traditional name of wind-f****r!). Sometimes instead it will swoop low over the ground to surprise prey.
It watches the ground below for small movements, then drops down onto some hapless creature. In Britain this is commonly a mouse, shrew or vole, while further south it is more likely to be a lizard or snake. Nor are they too proud to eat beetles, grasshoppers and other smaller prey.
I've occasionally seen them attack birds on the ground -- once a starling, and once I found one in a life-or-death struggle with a collared dove, which is more or less the same size. Unlike other falcons they do not seem to hunt in the air much.
Despite not being a large bird, a kestrel's eyes are about two-thirds as big as a human's, and each is roughly the size of its brain. Their vision is very sharp -- probably even sharper than a human's, especially at spotting small movements. Like other falcons they often bob their heads up and down to get an extra viewpoint, giving them a sort of "quadrocular" vision.